Heat the oil in a large frying pan, and add the chopped onion, minced garlic, grated carrot, grated courgette, and diced mushrooms. Cook over a medium heat for 5-10 minutes, stirring regularly, until all of the vegetables are completely soft.
Meanwhile, add the drained chickpeas to a food processor, and blitz for a few seconds until coarsely chopped.
When the vegetables are soft, add the tinned tomatoes, tomato puree, chopped chickpeas, oregano, and a generous pinch of salt and pepper. Bring to a simmer, and allow to cook for another 10 minutes or so, until the sauce is rich and fairly thick.
Cook the pasta in plenty of hot water until al dente (it will soften a little more in the oven, so don't overcook it at this stage). Drain, and then combine with the chickpea bolognese sauce. Mix well.
Transfer the mixture to a large baking dish or casserole dish, and top with the torn mozzarella, grated cheddar, and halved cherry tomatoes. Cover the dish, and bake at 190°C (Gas Mark 5 / 375°F) for around 25-30 minutes, until cooked to your liking. Serve with fresh basil to garnish, if desired.